Rolle's theorem

In calculus and real analysis, Rolle's theorem (or lemma) states that a real-valued differentiable function which attains equal values at two distinct points must have a stationary point somewhere between them, that is, a point where its derivative is zero. The theorem is named after Michel Rolle.

The theorem is a special case of, and is used to prove, the mean value theorem.

Statement

If a real function f is continuous on a proper closed interval [a, b], differentiable on the open interval (a, b), and f(a) = f(b), then there exists at least one c in the open interval (a, b) such that

f

′

(

c

)

=

0.

{\displaystyle f'(c)=0.}

History

In the 12th century Bhāskara II states an early form of Rolle's theorem in his work, though it was stated without a modern formal proof.

Although the theorem is named after Michel Rolle, Rolle's 1691 proof covered only the case of polynomial functions. His proof did not use the methods of differential calculus, which at that point in his life he considered to be fallacious. The theorem was first proved by Cauchy in 1823 as a corollary of a proof of the mean value theorem. The name "Rolle's theorem" was first used by Moritz Wilhelm Drobisch of Germany in 1834 and by Giusto Bellavitis of Italy in 1846.

Examples and counterexamples

Differentiability is not needed at the endpoints: Half circle

For a positive real number r, consider the function

f

:

[

−

r

,

r

]

→

R

{\displaystyle f:[-r,r]\to \mathbb {R} }

such that

f

(

x

)

=

r

2

−

x

2

{\displaystyle f(x)={\sqrt {r^{2}-x^{2}}}}

for all

x

∈

[

−

r

,

r

]

.